The Minimax Learning Rate of Normal and Ising Undirected Graphical Models

06/18/2018

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by   Luc Devroye, et al.

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Let G be an undirected graph with m edges and d vertices. We show that d-dimensional Ising models on G can be learned from n i.i.d. samples within expected total variation distance some constant factor of {1, √((m + d)/n)}, and that this rate is optimal. We show that the same rate holds for the class of d-dimensional multivariate normal undirected graphical models with respect to G. We also identify the optimal rate of {1, √(m/n)} for Ising models with no external magnetic field.